Let ${\lambda }_f(n)$ be the $n$th normalized Fourier coefficient of a holomorphic cusp form $f$ for the full modular group. In this paper, we established asymptotic formulae for high power sums of Fourier coefficients of cusp forms and further improved previous results. Moreover, as an application, we studied the signs of the sequences $\{{\lambda }_f(n)\}$ and $\{{\lambda }_f(n){\lambda }_g(n)\}$ in short intervals, and presented some quantitative results for the number of sign changes for $n\le x$.

Let $H_k^{\ast }$ denote the set of normalized primitive holomorphic Hecke cusp forms of even integral weight $k$ for the full modular group. Denote by ${\lambda }_{f\times f\times f}(n)$ the $n$th coefficient of the triple product $L$-function $L(f\times f\times f,s)$ attached to $f\in H_k^{\ast }$. Suppose $Q(\underset{\_}{x})$ is a primitive integral positive-definite binary quadratic form of fixed discriminant $D<0$ with the class number $h(D)=1$. In this paper, we study the distribution of ${\lambda }_{f\times f\times f}(n)$ on the set of all primes and its subset $\{p:p=Q(\underset{\_}{x})\}$ and obtain the analytic density and the natural density of the above sets. These results generalize previous ones.